302 10. EUCLIDEAN GEOMETRYFIGURE 18. Finding the surface area of a sphere.
that the area of the surface obtained by revolving the broken line ABCKL about
the axis AA! is ð AM • A'B. The method of exhaustion then shows that the product
AM • A'B can be made arbitrarily close to the square of AA'; it therefore gives the
following result (Proposition 33): The surface of any sphere is equal to four times
the greatest circle in it.
By the same method, using the inscribed right circular cone with the equatorial
circle of the sphere as a base, Archimedes shows that the volume of the sphere is
four times the volume of this cone. He then obtains the relations between the areas
and volumes of the sphere and circumscribed closed cylinder. He finishes this first
treatise with results on the area and volume of a segment of a sphere, that is, the
portion of a sphere cut off by a plane. This argument is the only ancient proof of
the area and volume of a sphere that meets Euclidean standards of rigor.
Three remarks should be made on this proof. First, in view of the failure
of efforts to square the circle, it seems that the later Greek mathematicians had
two standard areas, the circle and the square. Archimedes expressed the area of
a sphere in terms of the area of a circle. Second, as we have seen, the volume
of a sphere was found in China several centuries after Archimedes' time, but the
justification for it involved intuitive principles such as Cavalieri's principle that do
not meet Euclidean standards. Third, Archimedes did not discover this theorem
by Euclidean methods. He told how he came to discover it in his Method-
ic. The Method. Early in the twentieth century the historian of mathematics
J. L. Heiberg, reading in a bibliographical journal of 1899 the account of the dis-
covery of a tenth-century manuscript with mathematical content, deduced from a
few quotations that the manuscript was a copy^20 of a work of Archimedes. In
(^20) A copy, not Archimedes' own words, since it was written in the Attic dialect, while Archimedes
wrote in Doric. It is interesting that in the statement of his first theorem Archimedes refers to a
"section of a right-angled cone ÁÂÃ," and then immediately in the proof says, "since ABT is a